Abstract This memoir develops, discusses and compares a range of com- mutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smal- ler dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general - any compact set which is the closure of its interior - while in the second half we concentrate on the so-called canonical patterns. The topological invariants used are vari- ous forms of K-theory and cohomology applied to a variety of both C*-algebras and dynamical systems derived from such a pattern. The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or self-similarity, but one of the main motivations is also to provide an accessible approach to the the KQ group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern. The main results provide complete descriptions of the (un- ordered) X-theory and cohomology of codimension 1 projection pat- terns, formulae for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canon- ical projection patterns. Computations are made for the Ammann- Kramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One co- rollary is that, generically, projection patterns cannot be derived via substitution systems. Mathematics Subject Classification Primary: 52C23 Mathematical Subject Classification Secondary: 19E20, 37Bxx, 46Lxx, 55Txx, 82D25 ix
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